Question

find the domain of the function.
$f(x)=\frac{3}{x^{2}+9}-\frac{6}{x^{2}-16}$
what is the domain of f?
(type your answer in interval notation.)

Explanation:

Step1: Identify undefined points

A rational function is undefined when its denominator equals 0. For the first term, $x^2 + 9 = 0$ has no real solutions (since $x^2 \geq 0$, so $x^2 + 9 \geq 9 > 0$). For the second term, solve $x^2 - 16 = 0$.
$x^2 - 16 = 0 \implies x^2 = 16 \implies x = \pm 4$

Step2: Define valid x-values

All real numbers except $x=4$ and $x=-4$ are allowed.

Step3: Write in interval notation

Combine the intervals of valid x-values: $(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$

Answer:

$(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$